Fundamentals of Mathematics by Harry McLaughlin

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COURSE NOTES

FUNDAMENTALS OF MATHEMATICS

Spring 2006

Harry W. McLaughlin

August 25, 2006


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Copyright c2006 by Harry W. McLaughlinAll rights reserved.


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PREFACE

The main takeaway for participants in the course Fundamentals ofMathematics is an enhanced skill in writing mathematical proofs.

Most of the participants efforts are directed toward worksheets andhomework problems. Classroom time is devoted to the worksheets;there are no lectures.

The author believes that sharpened skills result from an aggressive

attack on the homework problems. Experience shows that completingthe homework problems requires substantial time set aside for idea-sorting; success usually means coming to grips with the worksheetsfirst.

The worksheets are available in an accompanying document. Solutionsto the worksheets discussed during the class meetings are distributedat the ends of the discussions.

These notes have been written to support the homework efforts. Theyare not intended as a text. They provide needed definitions and once

in a while reflect the instructors bias on how to say it.


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Contents

1 INTRODUCTION 1

1.1 These Notes . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Commonly Used Symbols . . . . . . . . . . . . . . . . . 2

2 SETS 3

2.1 Abstract Sets . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Unions, Intersections and Complements . . . . . . . . . 8

2.3 Existential and Universal Quantifiers . . . . . . . . . . . 11

2.4 Special Sets With Notation . . . . . . . . . . . . . . . . 12

3 FUNCTIONS 133.1 General Definitions . . . . . . . . . . . . . . . . . . . . . 14

3.2 Composition and Inverse Functions . . . . . . . . . . . . 18

3.3 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . 20

4 THE REAL NUMBERS 21

4.1 How To Think About Them . . . . . . . . . . . . . . . . 21

4.2 Fields and Ordered Fields . . . . . . . . . . . . . . . . . 22

4.3 Least Upper Bound Property . . . . . . . . . . . . . . . 24

4.4 Definition of Real Numbers and Integers . . . . . . . . . 25

4.5 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.6 Rational and Irrational Numbers . . . . . . . . . . . . . 28

4.7 Sequences of Real Numbers . . . . . . . . . . . . . . . . 28

4.8 Real-valued Functions on R . . . . . . . . . . . . . . . . 32

5 CARDINALITY 33

5.1 Arbitrary Sets . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Subsets of the Real Numbers . . . . . . . . . . . . . . . 37

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vi CONTENTS

6 INDUCTION 396.1 Axiom or Theorem? . . . . . . . . . . . . . . . . . . . . 396.2 Uses of the Induction Theorem . . . . . . . . . . . . . . 40

7 DECIMAL REPRESENTATIONS OF REAL NUM-BERS 437.1 Decimal Representations of Real Numbers . . . . . . . . 437.2 Ternary Representations and the Cantor Set . . . . . . . 48

8 EUCLIDEAN SPACES 51

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 518.2 R3 and Rn. . . . . . . . . . . . . . . . . . . . . . . . . . 558.3 Vectors or Points? . . . . . . . . . . . . . . . . . . . . . 578.4 Topology ofR2 . . . . . . . . . . . . . . . . . . . . . . . 588.5 Functions as Vectors and More . . . . . . . . . . . . . . 588.6 Lines, Chords, and Half Spaces . . . . . . . . . . . . . . 598.7 Supporting Lines . . . . . . . . . . . . . . . . . . . . . . 628.8 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

9 ABSTRACT VECTOR SPACES 679.1 Definition of a Vector Space . . . . . . . . . . . . . . . . 679.2 Definition of an Inner Product . . . . . . . . . . . . . . 69

9.3 Definition of a Norm . . . . . . . . . . . . . . . . . . . . 70

10 METRIC SPACES 7310.1 Definition of a Metric . . . . . . . . . . . . . . . . . . . 7310.2 S equences . . . . . . . . . . . . . . . . . . . . . . . . . . 7410.3 Topology of Metric Spaces . . . . . . . . . . . . . . . . . 7610.4 Interior and Boundary Points . . . . . . . . . . . . . . . 7710.5 Subspaces and Their Topologies . . . . . . . . . . . . . . 7710.6 Continuous Functions . . . . . . . . . . . . . . . . . . . 78


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Chapter 1

INTRODUCTION

1.1 These Notes

The course Fundamentals of Mathematics offers a chance for stu-dents of mathematics to develop a mind-set for careful mathematicalthought. It is intended as a precursor to upper level mathematicscourses where students ability to write careful mathematical argu-ments is assumed. Even though the course is called Fundamentals of

Mathematics its purpose is to provide a skill-developing opportunityand only in a secondary way, a knowledge enhancement opportunity.

One finds, here, mostly definitions and a few examples. Whats miss-ing are paragraphs explaining why history has filtered out the includedtopics. This leaves the door wide open for course instructors to paintpictures corresponding to their own world-views.

One can argue that logic and set theory are big pieces of the fun-damentals of mathematics. However, the writer believes that many(most) mathematicians are able to contribute to the state-of-the-artof their particular disciplines without chasing their arguments back tothe fundamentals. And in fact students are able to gain skill in writ-ing mathematical proofs without having spent a large amount of timestudying logic and set theory.

With this understanding these notes develop logic and set-theoreticconstructs only to the extent that they are needed for developing the-orem proving skills.

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2 CHAPTER 1. INTRODUCTION

1.2 Commonly Used Symbols

Logic and set-theoretic symbols are used throughout the notes. Mostof them are defined at the point of first use. A short list is providedhere along with the English words used to read them.

and or not implies if and only if logically equivalent union intersection element of empty set subset of there exists

for all

In the axioms for a field of Chapter #4 one finds the axiom

(x F) ((x) F) x+(x) = 0 (additive inverse).This is read: For all (for each) x in the set Fthere existsan element labelled, x, such that x + (x) = 0.

It is noted that the symbol is also used in the expressionf : A B in denoting a function from the set A to the setB.

The axioms of set theory and truth tables, found in the study of logic,are being swept under the rug. For sure, complete understanding ofthe fundamentals needs an in-depth investigation of the axioms of settheory and the basics of propositional logic. However, it is possible tosharpen skills in theorem proving, by absorbing the culture of theliterature. That is the tack taken here; these notes offer a glimpse ofthe culture.


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Chapter 2

SETS

The theory of sets exists as a foundational mathematical discipline;it supports virtually all of mathematics. The Zermelo-Fraenkel settheory is well developed and widely accepted; it is built on a handfulof basic symbols and axioms, but its power is far reaching. Interest-ingly, most practicing mathematicians are able to push the frontiersof mathematics without being conscious of how their work interfaceswith set theory. (One can successfully drive an automobile withouthaving a look under the hood.)

To explore set theory in detail would lead us away from the focus ofthese notes, namely a very brief introduction to classical analysis andvector spaces. So, in this chapter, we explore just enough set theoryto uncover its foundational role.

2.1 Abstract Sets

The mathematical construct called a set underlies most of mathemat-ics. With regard to these notes, the real numbers, vector spaces, metric

spaces and functions are all sets. Indeed, it is difficult to find a math-ematical construct that is independent of the notion of a set.

We first address the question:

To what mathematical construct does the term set refer?

At the interface between the spoken language and the language ofmathematics the term set is used to name a mathematical expression;

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4 CHAPTER 2. SETS

the expression usually models what humans perceive as a collection ofobjects. The term element of a set is used to name a mathematicalexpression that models one of the objects.

However, within the language of mathematics, the terms set and el-ement of a set, have context free mathematical roles, ones that existindependently of the notion of a collection of objects. Before goingthere we think more about modeling a collection of objects.

For illustration, we provide a mathematical model of the English sen-

tence Xavier lives in Seattle. It is,

x S.Here the symbol x models Xavier and the symbol models belongsto. The symbol S models the set of people who live in Seattle.

On the other hand, it is not necessary to start with:

Xavier and the set of people who live in Seattle.

One can start with the context free mathematical expression

x S,which our mathematics admits as being syntactically correct.1 (It neednot model Xavier lives in Seattle.)

The expression x S is admitted without a spoken language descriptioneven though, in human-human communication we frequently say thatx denotes an element of the set, S, or the element x belongs to theset, S.

In this treatise we follow common practice and write e.g. Let S de-note a set. or Let x denote an element of the set, S. In the first

case we mean that the mathematical expression x S is syntacticallycorrect, and in the second case we mean that the mathematical ex-pression x S is assigned the value true in the discussion at hand.

The above paragraphs crystallize the idea behind our definitions ofsetand element of a set. They are given next.

1For example, the expression xS is not admitted as syntactically correct.


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2.1. ABSTRACT SETS 5

A set is a symbol, here denoted by S, such that for somesymbol, x, the expression x S is syntactically correct.

An element of a set S is a symbol, here denoted by, x, forwhich the expression x S is assigned the value true.

It is noted that the definitions above consist of assigning words (phrases)from the spoken language to particular symbols: that both set and el-ement are defined to be symbols. In this regard the definitions above,

exist at the interface between mathematics and the spoken language.

To summarize: sometimes we assign, to a mathematical symbol or to agroup of mathematical symbols, the term set; sometimes we assign toa symbol the term elementof a set. These are declarations to enhanceunderstanding and communication. They are not needed for doingthe mathematics i.e., manipulating strings of symbols.

We take a slight detour. Roughly speaking, mathematicsconsists of (i) an alphabet, (ii) syntactically correct stringsof symbols from the alphabet, (iii) rules for combining thestringsthe rules preserve syntax correctness and (iv) a no-tion of truth.2 When humans communicate they assignspoken language phrases to these mathematical constructs.This is consistent with our belief, that

mathematical definitions link the spoken language to

mathematical constructs.

In the mathematical literature one finds the word defineused in an additional context. For example, in the nextparagraph we define the expression A B by A B (x)(x A x B). The intent is to use the expressionA B as shorthand for (x)(x A x B). In thatsense, A B is being notationally defined. In short: anotational definition allows two different symbol strings to

2The mathematical notion of truth can be thought of as a function that assignsto some syntactically correct strings, one of the values, true or false. The assignmentis done according to predefined rules of logic. This has the advantage that truth canbe determined by a computing machine and is independent of human judgement.


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6 CHAPTER 2. SETS

be used, one in place of the other. Common usage permitsone to write A B is defined by ... in place of A Bis notationally defined by ....3

In the mathematical literature one finds that the distinc-tion between a mathematical definition and a notationaldefinition is frequently blurred. This is common practiceto which we subscribe throughout.

In modeling human-perceived and human-imagined phenomena therefrequently arises a need for modeling a subset. A set A is a subset ofa set B if the expression (x)(x A x B) is assigned the valuetrue. To indicate that A is a subset of B one defines the expression,A B, by A B (x)(x A x B) and asserts that A B istrue.

It is is noted that the definition of the expression A B does notdepend upon the spoken language.

Again, in the realm of mathematical modeling, there sometimes arisesa need to model the human perceived notion that two sets are identical,that is, they contain exactly the same elements. The sets A and B areequal, written A = B, if the expression A = B is assigned the valuetrue. The expression A = B is defined by

(A = B) (A B B A).That is, A = B is true exactly when both A B and B A are true.

The definition of the equality symbol just given is an axiom of settheory.

The set A is said to be a proper subset of the set B if A

B and

A = B.The empty set denoted here by is a set defined (implicitly) by

(x)x .3The expression x is discussed later in this chapter; it is read for all x. The

symbol indicates logical equivalence. In the current use it indicates that A Bis true exactly when (x)(x A x B) is true.


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2.1. ABSTRACT SETS 7

In words, the empty set is a set that contains no elements, i.e. theexpression x is always assigned the value false. That the emptyset exists is an axiom of set theory. It can be argued that the emptyset is a subset of every set and that there exists at most one empty set.

The set product or Cartesian product of two non-empty sets, Aand B, is a set denoted by A B. It consists of all ordered pairs ofthe form (a, b), where a

A and b

B. One defines A

B by

(a, b) A B a A b B.

The footnote occurring at the end of this paragraph discusses the le-gitimacy of denoting an ordered pair by (a, b), but is not necessaryreading during an initial investigation of set products.4

Equality between two elements of A B, depends upon an assumednotion of equality in A and an assumed notion of equality in B. Twoelements of A

B, denoted by (a, b) and (c, d), are equal if a = c

and b = d. It is noted that if a A and b B then, in general,(a, b) = (b, a).

If the two sets A and B are identical then the set product of A andB is sometimes written A2. In particular, for the real numbers, R,(introduced below)5 one writes the set product ofR with itself, as R2.

4In writing the symbols (a, b) to represent an ordered pair, one relies on thereader being able to deduce that, in the assumed ordering, a precedes b. Someauthors comment that this approach imbeds the definition of ordered pair ina particular notation. One way to define the ordered pair consisting of a and bthat circumvents these objections, is to define it as a set of two elements, each ofwhich is a set. One of the sets is represented by

{a

}and the other is represented

by {a, b}. Then an ordered pair can be denoted by {{a}, {a, b}}, or {{a, b}, {a}}.Here one thinks of representing the element a by the set {a} and representing theelement b, by the set {a, b}. However, the writer believes that this definition is notneeded. The process of identifying syntactically correct strings relies on the abilityof humans to distinguish left from right: to distinguish the difference between (a, b)and (b, a), and the difference between x S and S x, etc. A similar remarkapplies to up from down. This human ability is assumed in almost all forms ofwritten communication.

5The real numbers are the numbers we deal with every day.


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8 CHAPTER 2. SETS

The set R2 is frequently referred to as the Cartesian plane. The mostcommon geometric (visual) representation ofR2 is a piece of paper orcomputer screen with an overlaid set of orthogonal axes. (However,R2 exists as a mathematical construct independently of geometry.)

Sometimes a set is written explicitly in terms of its elements. For ex-ample, if x S and x = y whenever y S, then one writes S = {x}and says that S consists of the single element, x.6 Similarly, onewrites, S = {a, b} and says that S consists of the two elements, aand b. (The order in which a and b are written is not important, i.e.,

{a, b} = {b, a}.) In general, curly brackets enclose elements of a set.When a set contains too many elements to list separately, curly brack-ets can also used to indicate set membership. For example, in the setof real numbers, R, the set of elements, each of whose absolute valueis two or less, is denoted by the expression {x R: |x| 2}. Theexpression is read: the set of real numbers, x, such that the absolutevalue of x is equal to or less than two.

2.2 Unions, Intersections and Complements

The set of all points belonging to either one of two sets is called theunion of the two sets. Specifically, let A and B denote sets. The unionof A and B, is a set. It is denoted by A B, and is defined by

A B = {x : x A x B}.

We read the right hand side of the above equality as the set of allelements each of which belongs to either the set A or the set B. Itis sometimes read asthe set of all x such that x belongs to A or xbelongs to B. The whole expression is shorthand for

x A B (x A x B).

(Recall that the symbol , in this case, means that the two math-ematical statements x A B and (x A x B) are logically

6It is noted that the expression {x} denotes a set, so the expression, x {x} issyntactically correct; it is assigned the value true.


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2.2. UNIONS, INTERSECTIONS AND COMPLEMENTS 9

equivalent, that is, for each choice of x, A and B they are either bothtrue or both false.

Similarly the set of all points belonging to both of two sets is calledthe intersection of the two sets. Specifically, let A and B denote sets.The intersection of A and B is a set. It is denoted by A B and isdefined by

A B = {x : x A x B}.We read the right hand side of the above equality as the set of all

elements each of which belongs to A and to B. This is shorthand for

x A B (x A x B).

The set of all points belonging to a parent set but not belonging toa particular subset is called the complement of the particular subset.Specifically, let A denote a subset of a set, U. The complement of A(in U) is a subset of U. It is denoted by Ac and is defined by

Ac = {x U : x A}.

In the event that the set A is not necessarily (wholly) contained in the

set U one can still speak of the complement of A in U. It is writtenU\A and defined by the same equation as above

U\A = {x U : x A}.

This expression can be used independently of whether or not A U.

Two sets, A, and B are disjoint if A B = .

The symmetric difference of two sets, A and B is denoted by ABand defined by

A

B = (A

B)

\(A

B).

One can show that AB = (A\B) (B\A). In words, the symmetricdifference of A and B is the set consisting of all points that belong toeither A or B but not both.

Before going on it is noted that the expressions A B, A B, x A, Ac, A\B and AB are not primitive symbols in set theory. Theyare abbreviations of expression involving only the primitive symbols


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10 CHAPTER 2. SETS

of set theory. However, these abbreviations are widely used; we usethem throughout.7

Sample Theorems

Theorem I. Let A, B and C denote sets. Then

(A B) C = A (B C).

Proof. It suffices to show that

(A

B)

C

A

(B

C)

and

A (B C) (A B) C.First, let x (AB)C. Thus, x (AB) and x C. Further, since x (AB),it follows that x A and x B. Thus, x A and x B and x C. Since x Band x C, it follows that x B C. Since x A and x B C, it follows thatx A (B C). One concludes that

(A B) C A (B C).

Next, let x A (B C). Thus x A and x B C. Further, since x (B C),it follows that x B and x C. Thus, x A and x B and x C. Since x Aand x B, it follows that x A B. Since x A B and x C, it follows thatx (A B) C. One concludes that

A (B C) (A B) C.This completes the proof.

Theorem II. (DeMorgans Law) Let A and B denote subsets of the set, U. Then

(A B)c = Ac Bc.


(A B)c Ac Bcand

Ac

Bc

(A B)c

.

First, let x (A B)c. Thus x A B. This implies that x A and x B. Thusx Ac and x Bc. Consequently, x Ac Bc. One concludes that

(A B)c Ac Bc.7We are resisting the temptation to dig deeper and start from scratch with the

primitive symbols of set theory.


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2.3. EXISTENTIAL AND UNIVERSAL QUANTIFIERS 11

Next, let x Ac Bc. Thus x Ac and x Bc. This implies that x A andx B; hence x A B. This means that x (A B)c. One concludes that

Ac Bc (A B)c.

This completes the proof.

Theorem III. (DeMorgans Law) Let A and B denote subsets of the set, U. Then

(A

B)c = Ac

Bc

.


(A B)c Ac Bcand

Ac Bc (A B)c.

First, let x (A B)c. Thus x A B. This implies that either x A or x B.If x A then x Ac. Consequently, x Ac Bc. If x B then x Bc and again,x Ac Bc. One concludes that

(A B)c Ac Bc.

Next, let x Ac Bc. Then x Ac or x Bc. If x Ac, then x A; hencex A B. This means that x (A B)c. If x Bc, then x B; hence x A B.This means that x (A B)c. One concludes that

Ac Bc (A B)c.


2.3 Existential and Universal Quantifiers

Above we introduced symbols for union, intersection, complement, el-ement of, and subset. There are two additional symbols of set theory:the universal and existential quantifiers. They are introduced below.

The existential quantifier is a symbol of set theory, written . Itis used to model the words there exists. For example, the expres-sion, x R : x2 = 2, is read there exists a real number x such


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12 CHAPTER 2. SETS

that the square of x equals two; that is, the expression is used toassert the existence of the square root of two. Equivalently, one some-times writes the expression (x R) x2 = 2 or as (x)(x Rx2 = 2).

The universal quantifier is a symbol of set theory, written, . Itis used to model the words for all. For example the expression,x2 0, x R, is read the square of x is greater than or equal tozero for all real numbers, x. Equivalently, one sometimes writes theexpression (x R) x2 0.

Note. With respect to the writing of the two quantifiers above oneis most likely to find the two expressions x R : x2 = 2 andx2 0, x R in the analysis literature and the two expressions(x)(x R x2 = 2) and (x R)x2 0 in the logic and set theoryliterature.

2.4 Special Sets With Notation

Following is a list of special subsets of the real numbers along withcommonly used notation. It is assumed that the reader has enough

understanding of these sets to allow them to be used in examples ofthis and the following chapter. Careful definitions of the sets comelater (in the chapter devoted to the definition of the real numbers).

The set of real numbers is denoted by R. The set of positive real numbers (not including zero) is denoted

by R+.

The set of integers (negative, zero and positive) is denoted by Z. The set of positive integers (not including zero) is denoted byZ+

.

The set of rational numbers is denoted by Q.


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Chapter 3

FUNCTIONS

In some of the examples of this chapter we use the symbol R to denotethe real numbers, even though a definition of the real numbers appearsin a subsequent chapter. We rely on the readers willingness to acceptthem here. In addition some of the examples include symbols for in-tervals; they too are introduced in a subsequent chapter. However, welist them here.

Special subsets of the real line, called intervals, are listed below. In

the list it is understood that < a < b < . Intervals are thosesubsets of the forms:

{x R : a x b}, denoted by [a, b]

{x R : a x < b}, denoted by [a, b)

{x R : a < x b}, denoted by (a, b]

{x R : a < x}, denoted by (a, )

{x R : a x}, denoted by [a, )

{x R : x < b}, denoted by (, b)

{x R : x b}, denoted by (, b]

{x R}, denoted by (, ).

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14 CHAPTER 3. FUNCTIONS

3.1 General Definitions

For two non-empty sets A and B, a function from A into B, denotedhere by f, is a subset of A B with the property: for each a A,there is exactly one b B such that (a, b) f. (Sometimes the wordsfrom A to B are used in place of the words from A into B.)

It is noted that the term function has been defined in terms of theprimitive term set. (A subset is a set.)

A function pairs each element, a A, with its own element b B;all such pairings are frequently denoted by a single symbol, for exam-ple, f. The pairing of a particular element a A with an elementb B is most often written as b = f(a). That is, one writes b = f(a)when (a, b) belongs to a function, f.

Some writers define a function, f, as a rule that assigns to each ele-ment a A an element b B. Still other writers define a functionfrom A into B as a mappingfrom A into B and denote such a mappingby f.1 We like this thinking and use it behind the scenes but dontadmit it into our mathematics.

In this document, as in much of the literature, the expression

f : A Bis used to denote a function from A into B. That is, the single symbol,f represents a whole subset of A B, namely, {(a, f(a)) : a A}.

It is noted that for a given function, f, each a A plays a role. Thatis f(a) is defined for each a A. However, it is not true, in general,that for a given function, f, each b B plays a role. In general,there may be many elements of B that do not have the form f(a) forsome a

A. As an example, the function f : R

R, defined by

f(x) = |x|, x R,2 doesnt see the negative real numbers. For1The words rule and mapping have intuitive appeal and are useful because they

guide thinking about functions. But they are not precisely defined mathematicalentities. So, saying, e.g., that a function is a rule or a function is a mapping,replaces the undefined word, function, with the undefined word rule or with theundefined word mapping.

2The expression f(x) = |x|, x R is a commonly used substitution for(x, f(x)) f, x f.


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3.1. GENERAL DEFINITIONS 15

example, there is no x R such that f(x) = 1, i.e., there is no x Rsuch that (x, 1) f.

Finally, we note that, in the context of functions from R into R, thesymbols f and f(x) have been used to represent two different concepts.The former denotes a function and the latter denotes a real number.Such a distinction is not always made in the literature. Sometimes,the symbol f(x) is used to denote a function and sometimes, to denotea real number, and sometimes, to denote both a function and a realnumber simultaneously: the choice depends upon the context.

For the function, f : A B, the domain of f, is the set A. Thecodomain is the set B. An element a in the domain of f is called anindependent variable (since it can be selected arbitrarily, and thenmapped by the function.)

For the function f : A B, the range of f is defined to be{b B : b = f(a), a A}.

An element b in the range of a function is called a dependent vari-able. (It is dependent in the sense that it is determined by the inde-

pendent variable once the function has been selected.)

It is noted that the range of a function, f : A B, is usually a propersubset of the set B.

Let f : A B and let a A. An element b B is sometimes calledthe image of a under f, if (a, b) f, i.e., if b = f(a). So the range ofthe function f is just its set of images. Sometimes the (whole) range iscalled the image ofA under f. In general, ifS A, one writes the im-age ofS under f as f(S) and defines it by f(S) = {(a, f(a)) SB}.

The above definitions of the wordimage

are worthy of a second look.Under f the image of a point is a point. But, under f the image ofa set is a set. So, because of the abuse of notation introduced above,one finds that f({a}) = {f(a)} and that f(a) = f({a}).

Let f : A B and let V B. The preimage ofV under f is denotedby f1(V) and defined by f1(V) = {a A : f(a) V}. Again thenotation needs a careful look. The expression f1(V) is defined for all


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subsets, V, of B but, in general, the expression f1(b) is not definedfor an arbitrary point b B. (The notion of an inverse function isdefined below, where, for some functions, f, the expression, f1(b), isdefined.)

If f : A B and the range off is identical to B, then the function issaid to be an onto function.3 An onto function is sometimes denotedby f : A

onto B.

It is noted that if f : [0, 1] [0, 1] and g : [0, 1] R andf(x) = g(x) = x

2

, x [0, 1], then f is an onto functionand g is not.

If f : A B, and if for each b belonging to the range of f, thereexists a unique a A such that b = f(a), then the function is saidto be one-to-one (sometimes written 1-1). A one-to-one function is

sometimes denoted by f : A1-1 B.

The function f : [1, 1] R, defined by f(x) = x3, x [1, 1] is one-to-one. The function g : [1, 1] R, definedby g(x) = x

2

, x [1, 1], is not one-to-one.

Let A denote a non-empty set. The identity function defined on A,denoted by I, is such that I : A A and I(a) = a, a A.

Two sets A and B are said to be in one-to-one correspondence ifthere exists f : A B that is one-to-one and onto.

The two functions, f and g, with f : A B and g : C D are saidto be equal if all three of the following hold: A = C and B = D andf(a) = g(a),

a

A.

The notion of equality of functions sometimes trips upmathematical writers. For example, let Z+ denote thepositive integers. Let f : Z+ Z+ and g : Z+ R,each be defined by f(n) = g(n) = n, n Z+. Then, eventhough f and g perform the same mapping, they are not

3Sometimes one says that the function maps A onto B.


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3.1. GENERAL DEFINITIONS 17

equal functions, since the domain of the function f is Z+and the domain of the function g is R. In particular, f isan onto function and g is not.

When a function, f : A B is defined as a rule or a mapping, thereis a need to define what is commonly called a graph of the function,f. It is usually defined as {(a, f(a)) A B}. Since the term graphappears so widely in the literature, it is defined separately here, even

though it is identical to our definition of a function.

The graph of a function from A into B is the function itself. In par-ticular, if f : A B, then the graph of f is {(a, f(a)) A B}.4

In order to motivate the next definition we get ahead of ourselves.Some calculus constructs, such as continuity and differentiability, arelocal; they deal with the behavior of a real-valued function definedin a neighborhood of a particular point in R. Characteristics of thefunction at points outside of a neighborhood5 of the particular pointare not relevant. Sometimes our definition of a function gets in the

way when investigating local constructs. We illustrate this with anexample. Let f : [1, 1] R be modeled by f(x) = x2, x [1, 1]and let g : R R be modeled by g(x) = x2, x R. Here f and gare two different functions since they have different domains. Howeverin the neighborhood (1, 1) of 0 they cannot be distinguished, one-from-another. There is a need for a mathematical construct that willallow us to transfer local properties of f to g and vice-versa. Such aconstruct is called a function restriction.

Let A, B and C denote non-empty sets with A B. Let f : B C.Let g : A

C be such that g(x) = f(x),

x

A. Then g is called

the restriction off to A. Later we use the notation fA to denote therestriction of f to A.

4That is, the graph of a function from A into B is precisely the subset of A B,that is used to define the function. For real-valued functions defined on intervals,it is commonly called a curve in the plane.

5We delay until later a precise definition of the technical term neighborhood.


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3.2 Composition and Inverse Functions

We define the notion of the composition of two functions. Let A , B , C , Ddenote non-empty sets and let f : A B and g : C D. If the rangeof g is a subset of A then the composition of f with g, denoted hereby f g, is the function, f g : C B, that is defined by

f g = {(c, f(g(x))) : c C}.

This is frequently written, f g(c) = f(g(c)), c C.6

Example. Let f : R R be defined by f(x) = x2, x Rand g : R R be defined by g(x) = x3, x R. Thenthe composition function, f g : R R is defined byf g(x) = x6, x R.

Example. Let g : R R be defined by g(x) = x2+ 1, x R and let f : [0, ) R be defined by f(x) = x, x [0, ). Then f g : R R is defined by f g(x) =

x2 + 1, x R.

It is noted that g f : [0, ) R and is defined byg f(x) = (x)2 + 1 = x + 1, x [0, ).

For this example, two observations are important:

f g = g f and even though the function prescribed by x+1 is defined

on R, the function g f is only defined on [0, ).

The notion of an inverse function plays an important role in mathe-matics. For example, the natural logarithm function can be definedas the inverse of the exponential function and vice versa. The inversecosine function plays a central role in defining the notion of an anglebetween two vectors.

6Since f g denotes a single function, one denotes the value of that function atthe point c by f g(c). It could be written (f g)(c) but the extra parentheses aresuperfluous.


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3.2. COMPOSITION AND INVERSE FUNCTIONS 19

A formal definition of an inverse function is given next.

Let A and B denote nonempty sets and let f : A11 B. Let rangef

denote the range of f. The function, denoted here by f1, such thatf1 : rangef A and defined by f1 = {(b, a) : (a, b) f} is calledthe inverse function for f (or the inverse function of f).

First, it is noted that f1 is indeed a function. And further, it is notedthat f1 is 1-1 and onto.7

Example. Let f : R R be defined by f(x) = x3, x R.Then the inverse function, f1 : R R is defined byf1(x) = x1/3, x R. It is noted that in this case, thefunction, f, is both one-to-one and onto.

Using the notion of function composition it is noted that the function,denoted here by f1 f, such that f1 f : A A, and defined byf1 f(a) = a, a A, is equal to the identity function on A. Thatis, the composition of f1 with f is equal to I.

Let f : Z Z be defined by f(n) = n + 1, n Z. Thenf1 : Z Z is defined by f1(n) = n1, n Z. Finallyf1 f is the identity function on Z.

There is some careful notation bookkeeping needed when dealing withinverse functions. This is illustrated in the following example.

Let f : [0, ) [0, ), be defined by f(x) = x2, x [0,

). Since f is one-to-one and onto, it has an inverse

function defined on [0, ). What does it look like? Solv-ing the equation, y = x2, for x, one obtains x = y1/2, y [0, ). That is, f1 : [0, ) [0, ) and is defined byf1(y) = y1/2, y [0, ). When it comes time to plotthe graph of f1 on an x, y coordinate system one needs

7There is an anomaly here. Both f and f1 are 1-1 functions. But f1 is alwaysan onto function even when f is not an onto function.


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to take some care. For it is customary to label the hori-zontal axis with an x and let the appropriate points on theaxis represent values in the domain of the function to beplotted. But above we labelled the elements in the domainof f1 with the symbol y. So there are two alternatives:(1) the symbol y needs to be replaced with the symbolx and the symbol x replaced with the symbol y, in thefunction form of f1, or (2) the labels on the axis of theplot need to be changed. It is customary to accept alter-native (1). In this case the function form of f1 becomes

f1(x) = x1/2, x [0, ). Having done this, one cannow plot the graphs of both of the functions, f and f1 onthe same coordinate system (and note that they are sym-metric about the plot of the graph of the line modeled byy = x, x [0, )).

3.3 The Axiom of Choice

The axiom of choice is an axiom of set theory. Whole books have beenwritten on this one axiom; of interest are the non-intuitive results thatemerge as its consequences. One of the most famous is the so-called

Banach-Tarski paradox(1924). Banach and Tarski proved that one can(mathematically) take apart a 3-d sphere in a finite number of piecesand then put the pieces back together in another way, resulting in twospheres each identical to the original. (It is known that the minimumnumber of pieces for which this is possible is five.) We dont investi-gate this paradox here, but give a precise statement of the axiom ofchoice.

Axiom of Choice Let C denote a collection of non-empty subsetsof a set K. Then there exists a function f : C K such thatf(S) S, S C.

Roughly speaking the axiom of choice says that it is possible to forma set by selecting one element form each set belonging to a collectionof sets.


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Chapter 4

THE REAL NUMBERS

Below we define the real numbers as an ordered field with the least up-per bound property. The notions of field, ordered field and least upperbound property are defined in this chapter. But before going there wediscuss briefly the notion of a real number.

4.1 How To Think About Them

The real numbers are the numbers that are used every day by theperson on the street, as well as those using mathematics in relateddisciplines. They come in the form (i) rational numbers, e.g., 3 and4

5, and (ii) irrational numbers, e.g., 2 and .1

The real numbers reside in the imagination of humans. We are notable to detect them with our five senses. No one has ever seen one or

1One could argue that there are way too many real numbers for any one of usto deal with in a lifetime; since our lives consist of a finite number of days andeach day we can deal individually with only a finite number of real numbers. Infact, computers can represent, in decimal form, at most a finite number of numbers!

In spite of this, our real number system contains an infinite number of numbers, infact an uncountable number of them. The integers are clearly countable and witha little effort one can argue that the rational numbers are countable. However, theirrational numbers are not countable: they are uncountable. That means that ifwe think of all of the real numbers as residing in a bag, and we reach in to takeone out, the probability of selecting an integer or a rational number is zero! So,the probability of selecting one of the numbers that we deal with, in decimal form,during our lifetime, is zero.

21


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22 CHAPTER 4. THE REAL NUMBERS

tasted one. They exist in our minds as an abstraction.

The definition of the real numbers, given below, includes the com-monly accepted notions of addition and multiplication, as well as thecommonly accepted notion of order (one number being greater thananother). It also includes the technical (but important) notion of com-pleteness.2

4.2 Fields and Ordered FieldsA field is a non-empty set, F, along with two functions from F Finto F. The first is called addition. The image under the additionfunction of the element, (x, y) F F, is denoted here by x + y.The second function is called multiplication. The image under themultiplication function of the element, (x, y) F F, is denoted hereby xy. The two functions satisfy the following nine axioms:

1. (x F)(y F) x + y = y + x (additive commutative law)

2. (x F)(y F)(z F) (x + y) + z = x + (y + z) (additiveassociative law)

3. (0 F) (x F) x + 0 = x (additive identity)

4. (x F) ((x) F) x + (x) = 0 (additive inverse)

5. (x F)(y F) xy = yx (multiplicative commutative law)

6. (x F)(y F)(z F) (xy)z = x(yz) (multiplicative associativelaw)

7. (1 F) (x F) 1x = x (multiplicative identity)

8. (x F, x = 0) (x1 F) xx1 = 1 (multiplicative inverse)

9. (x F)(y F)(z F) x(y + z) = xy + xz. (distributive law)

NOTE. It is noted that:

(i) (0) = 0.2Roughly speaking, the real numbers are complete in the sense that, with respect

to their ordering and their metric structure, they contain no gaps.


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4.2. FIELDS AND ORDERED FIELDS 23

(ii) 11 = 1 (provided that F contains at least two distinct ele-ments).

(iii) The distributive law ensures a compatibility between the addi-tion and multiplication functions.

(iv) There exists a field consisting of exactly two elements, {0, 1},with addition and multiplication defined by: 0 + 0 = 0, 0 + 1 =1 + 0 = 1, 1 + 1 = 0 and 00 = 0, 01 = 10 = 0, 11 = 1. (All ofthe axioms are satisfied.)

The rational numbers (defined later) are a field and have a countablenumber of elements. The real numbers (defined later) are a field, buthave an uncountable number of elements.

A field Fis ordered if there is a non-empty subset ofF\{0}, denotedhere by F+ that satisfies the following three axioms:

1. (x F+)(y F+) x + y F+ (closed under addition)

2. (x F+

)(y F+

) xy F+

(closed under multiplication)

3. for each x F exactly one of the following is true: (i) x F+, (ii)x F+, (iii) x = 0. (trichotomy)

The set F+ is called the set ofpositive elements in an ordered field.We say that the field F is ordered by the subset F+.

Let x and y denote elements of an ordered field F. Using the expres-sion y x in lieu of y + (x), one writes x < y if y x F+. In thiscase one says that x is less than y and that y is greater than x. Onewrites x

y if either x = y or x < y. One writes y > x if x < y. One

writes y x if x y.

Theorem. Let F denote a field ordered by the subset F+ and letx F. Then x F+ if and only if x > 0.

The proof follows immediately from the identity x = x 0.


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4.3 Least Upper Bound Property

The least upper bound property defined here ensures that the realnumbers have no holes, that, e.g.

2 does indeed exist as a real num-

ber.

Let A denote a subset of an ordered field, F. An element M F iscalled an upper bound for A if (x A) x M. A similar definitioncan be made for the notion of lower bound. A subset ofFthat hasan upper bound is said to be bounded above. A similar definition

is made for bounded below. A subset of F that is both boundedabove and bounded below is said to be bounded.

Let A denote a non-empty subset of an ordered field, F. If A isbounded above and if M is an upper bound for A such that M Mwhenever M is an upper bound for A, then M is called the least up-per bound for A. (It is noted that it is possible to show that therecannot be two distinct least upper bounds for A, so that we are ableto speak of the least upper bound for A.)

Let A denote a non-empty subset of an ordered field,

F. If A is

bounded below and if M is a lower bound for A such that M Mwhenever M is a lower bound for A, then M is called the greatestlower bound for A. (As above, we are able to speak of the greatestlower bound for A.))

An ordered field, F, has the least upper bound property ifFcon-tains a least upper bound for every non-empty subset that is boundedabove. (It is noted that the least upper bound may, or may not, belongto the bounded subset.)

The next theorem argues that an ordered field with the least upper

bound property also has a greatest lower bound property.

Theorem. Let Fdenote an ordered field with the least upper boundproperty and let A denote a non-empty subset of F. If A is boundedbelow then Fcontains a greatest lower bound for A.

Proof. Let A denote a non-empty subset of Fthat is bounded below


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4.4. DEFINITION OF REAL NUMBERS AND INTEGERS 25

and let B be the set of lower bounds of A. That is

B = {x F: x y, y A.}Since A is bounded below, it follows that B = .

Since every element of A is an upper bound for B it follows that Fcontains a least upper bound for B; call it u. We argue that u is agreatest lower bound for A. It suffices to argue: (i) u y, y A,that is, u is a lower bound for A and (ii) if z > u for some z Fthenthere exists y

A such that y < z, that is, z is not a lower bound for A.

To argue (i) it suffices to note that ify A then y x, x B. Thatis, y is an upper bound for B. Since u is the least upper bound for Bit follows that u y.

To argue (ii) it suffices to note that if z > u then z B (else u is notan upper bound for B). Thus there exists y A such that y < z.


By the definition given below the real numbers have the least up-

per bound property. However the rational numbers, Q, do not, since{x Q : x2 < 2} is bounded above in the field Q but does not have aleast upper bound in Q.3

4.4 Definition of Real Numbers and Integers

The real numbers, denoted here by R, are an ordered field with theleast upper bound property.4 The positive real numbers are definedto be {x R : 0 < x}. The negative real numbers are defined to be{x R : x < 0}.

In order to define the integers we need the notion of the absolute-value function. That is given next. Let F denote an ordered field.

3To verify this one needs to argue that

2 is not a rational number. (Seethe Workbook.) This is an important result; it specifically identifies an irrationalnumber.

4We leave open the question of whether or not there exists at most one orderedfield with the least upper bound property.


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The absolute value function is a function from F into F. It isdenoted by |x|, x Fand defined for all x Fby

|x| =

x if x 0x if x < 0.

The workbook contains properties of the absolute-value function.

The set of integers is a subset ofR, denoted here by Z, that satisfiesthe following axioms:

(ii) 0 Z,

(ii) (n Z) ((n + 1) Z (n 1) Z),

(iii) (m, n Z) (m = n |m n| 1).

It can be shown that the set of positive integers, denoted here by Z+

is unbounded. (Similarly, the set of negative integers is unbounded.)

The following two theorems are workhorse theorems in characteriz-ing the integers. They follow from the least upper bound property; in

particular, they are used to prove the induction theorem.

Theorem. A non-empty set of integers, that is bounded below, con-tains its greatest lower bound. (Hence it contains a least element.)

Proof. Let U denote a non-empty set of integers that is bounded be-low. Let b denote the greatest lower bound, in R, ofU. We argue thatb U.

Since b + 1 is not a lower bound for U,5 there exists p U such that

b

p < b + 1.

We argue that p = b.

If q Z and q < p, then

q p 1 < b.5It can be argued that b + 1 > b.


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4.5. INTERVALS 27

Thus q U.

One concludes that p is the greatest lower bound for U. That is p = b.


Theorem. A non-empty set of integers, that is bounded above, con-tains a largest element.

4.5 Intervals

Special subsets of the real line, called intervals, are listed below. Inthe list it is understood that < a < b < . Intervals are thosesubsets of the forms:

{x R : a x b}, denoted by [a, b] {x R : a x < b}, denoted by [a, b) {x R : a < x b}, denoted by (a, b]

{x R : a < x}, denoted by (a, ) {x R : a x}, denoted by [a, ) {x R : x < b}, denoted by (, b) {x R : x b}, denoted by (, b] {x R}, denoted by (, ).

For the intervals [a, b], (a, b], [a, b), (a, ), [a, ), the real number a issaid to be a left end point of the interval. Similarly, for the inter-

vals, [a, b], (a, b], [a, b), (, b), (, b], the real number, b is saidto be a right end point of the interval.6 An element of an intervalthat is neither a left end point nor a right end point is said to be aninterior point of the interval. It is noted that every interval contains

6It is noted that is never an element of an interval, even though it appears inthe representations, (a, ) and [a, ). Additionally, is never an end point of aninterval, even though it appears in the representations, ( a, ) and [a, ). Similarstatements apply to .


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an infinite number of interior points.7

The closure of an interval I is denoted by I. It is I with its end pointsadjoined. For example if I = (a, b), then the closure of I is given byI = [a, b] and if I = (, b), the closure of I is given by I = (, b ],etc. Of course, the closure of [a, b] is itself.

4.6 Rational and Irrational Numbers

A real number is a rational number if it can be written in the formpq where p and q are integers with q = 0. A real number that is not arational number is an irrational number.8

It can be argued that the rational numbers are countable and the ir-rational numbers are uncountable. (See Chapter #5.)

It can also be argued that between any two distinct real numbers thereexist both a rational number and an irrational number. (See the Work-book.)

4.7 Sequences of Real Numbers

An infinite sequence in a set, S, is a function from Z+ into S.

The repeating decimal expansion for 37

, namely (writtenwithout the decimal point)

428571428571

is a representation of an infinite sequence in the set of ten

integers {0, 1, , 9}.Let n Z+. A finite sequence of length, n in a set, S, is a functionfrom the first n positive integers into S.9

7In particular, a single point is not an interval.8The expression p

qrepresents the product of p with the multiplicative inverse of

q.9The first n positive integers is the set given by {m Z+ : 1 m n}.


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4.7. SEQUENCES OF REAL NUMBERS 29

The binary expansion for the integer, 27, namely

1 1 0 1 1

is a finite sequence in the set {0, 1} (that is, in the setconsisting of two elements, 0 and 1).

An infinite sequence in a set, S, or a finite sequence in a set, S is asequence in a set S. (A sequence can be either infinite or finite.)

An infinite sequence in a setS

is frequently written {sk}

k=1 (ors1, s2,

)where it is understood that sk is the image of the positive integer, k,under the function from Z+ into S. A similar remark can be made fora finite sequence. In this case one writes {sk}nk=1 where it is under-stood that n Z+.

Sometimes it is more convenient to think of a sequence as a functionfrom the non-positive integers into S instead of as a function fromthe positive integers into S. If f : Z+ {0} S then the functiong : Z+ S defined by g(k) = f(k1), k Z+ is an infinite sequencein S. Since the infinite sequence g can be obtained from the functionf by a simple shift in the integers, we call the function f an infinite

sequence. For example we write {sk}k=0 and call it a sequence.

(We speak colloquially.) The sequence, {Sn}n=0, of par-tial sums of the geometric series, each defined by

Sn =

nk=0

rk, |r| < 1

form a sequence in R. For each n the value of Sn can bethought of as the image of n, under a function f : Z+ {0} R.

Following similar reasoning we write {sk}nk=0 and call it a finite se-quence.

Let {xn}n=1 denote a sequence in R and let x R. The sequence{xn}n=1 is said to converge to x, written

limn

xn = x


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if for all > 0 there exists N Z+ such that |xnx| < for all n > N.(It is understood that N may depend upon .) A sequence that con-verges to a real number x, is said to be a convergent sequence.

When the real sequence {xn}n=1 converges to the real number, x, thepoint x is said to be the limit of the sequence.

It can be argued that a real sequence can have at most one limit.

The sequence {1

n}

n=1 is a convergent sequence; it convergesto 0.

The infinite sequence, 1, 1, 1, 1, 1, 1, . is not a con-vergent sequence.

Let {xn}n=1 denote a sequence in R. The sequence {xn}n=1 is said toconverge to , written

limn

xn =

if M R there exists N Z+ such that xn > M for all n > N. (Itis understood that N may depend upon M.) (A similar definition isgiven for .)

The sequence in R given by {1, 2, 3, } converges to .It is noted that a sequence that converges to is not a convergentsequence.10

The sequence {xn}n=1 in R is said to be a Cauchy sequence if forall > 0 there exists N Z+ such that |xnxm| < for all m,n > N.(It is understood that N may depend upon .)

Using the least upper bound axiom it can be shown that if {xn}n=1 isa Cauchy sequence in R then there exists uniquely, x R such thatlimn xn = x.

10This seems like splitting hairs; it is. However, the phrase convergent sequenceis almost always applied to sequences that converge to real numbers.


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4.7. SEQUENCES OF REAL NUMBERS 31

The next definition shows that the notion of an infinite sum, eventhough it transcends the algebraic structure ofR, can be defined us-ing the notion of a sequence.

For the sequence {xk}k=1 in R, one denotes the nth partial sum ofthe sequence by Sn and defines it by

Sn =n

k=1

xk, n Z+.

The expressionn

k=1 xk needs careful definition; this is explored inChapter 6 of the Workbook.

If the sequence of partial sum, {Sn}n=1, converges to the real numberS, then one writes

k=1

xk = S.

One of the most useful infinite sums is the sum of thegeometric series. It is given by

k=0

rk =1

1 r,

r

(

1, 1).

Its partial sums are given by

Sn =

nk=0

rk =1 rn+1

1 r , r = 1, n = 0, 1,

The geometric series is explored in Chapter 6 of the Workbook.

Three workhorse theorems are given next. They are explored in Chap-ter 4 of the Workbook.

Theorem. If{xk}

k=1 and {yk}

k=1 are convergent sequences in R thenso is the sequence {xk + yk}k=1. Further

limk

(xk + yk) = limk

xk + limk

xk.

Theorem. If{xk}k=1 is a convergent sequence in R then so is {xk}k=1, R. Further

limk

xk = limk

xk.


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Theorem. For r R, r > 0 one has

limn

rn =

if r > 11 if r = 10 if 0 r < 1.

4.8 Real-valued Functions on R

Let D denote a non-empty subset ofR and let x0 D. Let f : D R.The function, f, is continuous at x0 if for every > 0 there ex-

ists > 0 such that |f(x) f(x0)| < whenever both x D and|x x0| < . (It is understood that may depend upon .)

In words, f is continuous at x0 if f(x) is close to f(x0) whenever bothx is in the domain D and x is close to x0.

The function f : R R defined by f(x) = x, x R iscontinuous at every point ofR.

Similarly, the function f : Z Z defined by f(z) =z,

z

Z is continuous at every point ofZ.

The function f : R R defined by

f(x) =

0 x 01 x > 0

is continuous at every point ofR except x = 0.

One can argue that the set of continuous real-valued functions eachdefined on a non-empty subset ofR is closed under addition and multi-plication. That is, the sum and product of two continuous real-valuedfunctions, each defined on the same set, are both continuous.

We leave untreated a full investigation of continuous real-valued func-tions. (See Chapter 10 for more discussion.)


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Chapter 5

CARDINALITY

5.1 Arbitrary Sets

A set is said to be finite if it is empty or if for some positive integer,n, there exists a 1-1 function from the first n positive integers onto theset.1 If for some positive integer, n, there exists a 1-1 function fromthe first n positive integers onto a set, S, the set, S, is said to havecardinality n: written |S| = n. The cardinality of the empty set is 0.

The set of all subsets of a set S is written 2S and called the powerset of S.

A theorem below states that a finite set has exactly one cardinality.This justifies use of the expression the cardinality of S instead of acardinality of S.

For a set S with cardinality n (n Z+) one can argue thatthe cardinality of the set of all subsets of S is 2n, i.e.,

|2S| = 2n.The right hand side of the above equation justifies the somewhat awk-ward notation, 2S, for the power set. (See the Workbook.)

A set is said to be infinite if it is not finite.

1The words the first n positive integers need definition. For n Z+ the firstn positive integers is the set given by {m Z+ : 1 m n}.

33


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34 CHAPTER 5. CARDINALITY

Theorem. A finite set has exactly one cardinality.

Proof. The proof is not provided here.

A non-empty set that can be put into 1-1 correspondence with a sub-set of the positive integers is said to be countable. (A countable setcan be either infinite or finite.)

The set of integers is countable.

The following theorem is oft used. Roughly speaking it says that acountable union of countable sets is countable. Sometimes it is phrasedas: A countable set of countable sets, is countable.

Theorem. Let {Sk}k=1 denote a sequence of countable sets. Thenk=1Sk is countable.2

Proof. The proof is left to the Workbook.

A non-empty countable set that is not finite is said to be countably

infinite.

A non-empty set that is not countable is said to be uncountable.

The real numbers constitute an uncountable set. (SeeChapter 7.)

The following paragraph deals with the question; Is there a largestset, in terms of the number of its elements? It also deals with thequestion of whether or not there is a set having more elements than

the positive integers but less elements than the real numbers?

The notion of cardinality of infinite sets is substantially more involvedthan the notion of cardinality for finite sets. Georg Cantor(1845-1918)spent a good bit of his life trying to uncover the relevant phenomena.

2The expression k=1Sk is a set; it is defined by k=1Sk = {x : x Sk for some k Z+}.


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5.1. ARBITRARY SETS 35

He said that two infinite sets have the same cardinality if there existsa 1-1 function from one onto the other. Otherwise they have differentcardinalities. To the integers, denoted by Z, he assigned the symbol0 and said that the integers have a cardinality 0. He then arguedthat the real numbers, R, do not have the same cardinality as Z.3 Hesaid that the real numbers have cardinality 1. He was able to provethat the set of all real-valued functions defined on R have still anothercardinality which he labelled 2. Finally he was able to prove that ifSdenotes any non-empty set, the set of all subsets of S, denoted by 2S,has greater cardinality than S.4 (His proof of this fact is, in the mind

of this writer, one of the most elegant proofs in all of mathematics.)Using this fact, Cantor was thus able to define an infinite sequenceof cardinal numbers, 0, 1, 2, . Here, for each i 1, i denotesthe cardinality of the set of all subsets of a set whose cardinality is i1.

Cantor was unable to prove the existence or lack of existence of a setwith cardinality greater than 0 and less than 1. He posited thatthere was no such set. This became known as the continuum hy-pothesis. In the 1960s Paul Cohen was able to prove, using themathematics we have at hand, that the continuum hypothesis is un-decidable.

One of the consequences of the work of Cantor and Cohen is that thetruth of the continuum hypothesis is independent of the set theorythat underlies almost all of commonly practiced mathematics. Of par-ticular concern is that we are not able to establish the truth of thestatement: A non-countable subset ofR can be put into one-to-onecorrespondence with R.

Finally we note an obvious theorem.

Theorem. Every subset of a countable set is countable.

Proof. The proof is not given here.

The following theorem, due to Cantor, and its cocktail-party-proof are

3His proof of this fact is a classic in mathematics. It uses the now-called Cantordiagonalization process.

4A set, A, is said to have greater cardinality than a set, B, if there exists afunction that maps A onto B but not vice-versa.


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offered as a starting point for a problem in the Workbook.

Theorem. Let S denote a non-empty abstract set. The cardinality of2S is different from the cardinality of S.

Cocktail-Party-Proof. The proof is by contradiction.

Assume that there exists a function, f : S 2S, that is both 1-1 andonto. If s S is such that s f(s), color s blue. Otherwise colors red. One notes that the preimage of the empty set is red and thepreimage of S is blue.

Let R denote the subset ofS that consists of exactly the red elements.Either R is the image, under f, of a blue element or a red element.

If R is the image of a blue element, then that blue element, by itsblueness must be in R, but this is contrary to the definition of R.Thus R is not the image of a blue element.

If R is the image of a red element, then that red element, by its red-

ness is not an element ofR, but this is also contrary to the definitionof R. Thus R is not the image of a red element.

One concludes that R is not in the range of f contrary to the assump-tion that f is onto.

This completes the proof.5

A formal proof of Cantors theorem is asked for in Chapter 5 of theWorkbook.

This result was what allowed Cantor to conclude that there exists aninfinite number of infinities. Such an unexpected result was not im-mediately accepted in the mathematical community.

5This argument can be found in The Colossal Book of Mathematics, by MartinGardner, Norton & Company, 2001, p.340.


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5.2. SUBSETS OF THE REAL NUMBERS 37

5.2 Subsets of the Real Numbers

Of significant interest is the fact that the rational numbers are count-able and the irrational numbers are uncountable.

In the proof of the following theorem there are two partially hiddensubtleties. First, the proof uses the fact that every non-empty subsetof a finite set is finite to conclude that a countable set containing acountably infinite subset must itself be countably infinite. Second, ituses the symbol pq where p and q are positive integers in two ways: (i)

as a formal symbol (In this use pq =p

q if and only ifp = p and q = q.)

(ii) as a representation of a rational number. (In this use pq =p

q if and

only if pq = qp.)

Theorem. The rational numbers in (0, 1] are countably infinite.

Proof. Every rational number in (0, 1] can be written in the form pqwhere p, q Z+, p q.

For each n

Z+, denote a set of formal symbols by Sn and define it

by

Sn =p

n: p Z+ and p n

, n Z+.

Clearly |Sn| = n, n Z+. Thus the set, S, of all formal symbolspq where p, q Z+, p q is a countable union of countable sets andhence countable.6

The rational numbers in (0, 1] can be put into 1-1 correspondence withsome subset of the set, S, namely the set of all formal symbols pq where

p and q contain no common integer divisors other than 1. Thereforethe rational numbers are countable.

7

It remains to argue that the set of rational numbers, Q, constitute aninfinite set. To this end one notes that the rational numbers in (0, 1]contain the infinite set {x Q : x = 1n for some n Z+} and therefore

6In fact it is a countable union of finite sets.7We have used the fact that every subset of a countable set is countable.


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constitute an infinite set.8

Theorem. The set of real numbers in (0, 1) is uncountable.

Proof. The proof is found in Chapter #7.

8We have used the fact that every subset of a finite set is finite. (See theWorkbook.)


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Chapter 6

INDUCTION

6.1 Axiom or Theorem?

Above we defined the real numbers axiomatically, the real numbersare an ordered field with the least upper bound property. There isanother approach to introducing the real numbers into mathematics.They can be constructed. The construction process starts with theso-called Peano Postulates, which are used to construct the positiveintegers. One of the postulates is the induction postulate. It says thatifn is an integer then it has an immediate successor, denoted by n + 1,which is also an integer.

However, we didnt introduce the real numbers this way, so we donthave the induction postulate in our tool box. Instead we hypothesizedthe existence of an ordered field with the least upper bound property.Then we hypothesized the existence of a subset of the field with certainproperties. We called the subset, the set of integers. We then used theleast upper bound property to argue that every set of positive integershas a least element. This is all that is needed to recover the inductiontheorem. The theorem is given next.

Induction Theorem. Let S denote a subset of the positive integerssuch that:

(i) 1 Sand(ii) if n S then n + 1 S.

39


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40 CHAPTER 6. INDUCTION

Then, S is the set of positive integers, i.e., S = Z+.

Proof. Since S Z+ it suffices to aruge that the complement of S inZ+ is the empty set. Let U = {p Z+ : p S}. We argue that U = .

The proof is by contradiction.

Assume U = . We argue that this implies that there exists q Z+such that

q

U and q

U

which contradicts one of the axioms of the underlying logic.1

Since U is a set of integers, bounded below by 0, it has a least element;call it q. In particular

q U.By (i) of the theorem hypothesis, q = 1. Thus, q 1 Z and in fact,using a result of the Workbook, one determines that q 1 Z+. Thusfrom (ii) of the hypothesis, (q 1) + 1 S. One concludes that

q = (q 1) + 1 U.

Thus both q U and q U.


6.2 Uses of the Induction Theorem

An important application of the Induction Theorem is in proving thatthe expression

nk=1 ak as we conventionally use it to designate the

sum ofn real numbers, is indeed well defined. One can start the argu-ment by saying that if

n1k=1 ak has been defined to be a real number

then one can definenk=1 ak by the equality nk=1 ak = n1k=1 ak + an.In this last equality the plus sign represent addition in the real num-bers. This almost looks like a complete argument, but it doesnt ruleout the possibility that for some n one is not able to define

n1k=1 ak.

The induction theorem rules out the possibility of this occurrence.This result is addressed in the Workbook.

1A logical proposition cannot be both true and false.


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6.2. USES OF THE INDUCTION THEOREM 41

Theorem. If 0 r < 1, thenk=0

rk =1

1 r .

Proof. We use the identityn

k=0 rk = 1r

n+1

1r , n Z+.2 Rewritingthis and using the fact that limn r

n+1 = 0, one obtains

k=0

rk = limn

n

k=0

rk

= limn

1

1 r 1

1 r rn+1

= limn

1

1 r 1

1 r limnrn+1

=1

1 r .


2This is verified using the induction theorem. See the Workbook.


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42 CHAPTER 6. INDUCTION


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Chapter 7

DECIMAL

REPRESENTATIONS OF

REAL NUMBERS

This chapter is devoted to decimal representations of real numbers.In using real numbers for mathematical modeling, the real numbersare almost always represented by decimal representations (base ten orother bases). In that regard the topic has been given chapter-status.

7.1 Decimal Representations of Real Numbers

Lemma. Let x R and let N Z+. Then there exists exactly oneinteger, n, such that

n

N x < n + 1

N.

Proof. The set of all integers, m, such that m

N x, is boundedabove so it contains a largest element. Call it n. Since n N x andn + 1 N x it follows that n + 1 > N x. Since N > 0 it follows that

n

N x < n + 1

N.

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44CHAPTER 7. DECIMAL REPRESENTATIONS OF REAL NUMBERS

To argue that there is exactly one such integer, n, one supposes thatm is an integer such that m = n and mN x. Then m N x. From theway n was selected above it follows that m < n and hence m + 1 n.Thus m+1N nN x, that is, x < m+1N .


Decimal Representation Theorem. Let 0 x < 1 and let n Z+.Then there exists a unique sequence

{ak

}k=1 in

{0, 1,

, 9

}such that

0 x n

k=1

ak

10k

0, the setS

D

r is wholly contained in one of the closed halfspaces determined by L. Such a line is called a local supporting line.

The next paragraph is entered with the understanding that the readerhas a familiarity with tangent lines.

It is noted that a tangent line to a graph may locally support the graphat the point of tangency. However, this is not always the case.


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8.8. ANGLES 63

For the function, f : R R, defined by f(x) = x3, x R,the line in R2 modeled by y = 0, is the tangent line to thegraph of f at (0, 0), but it does not locally support thegraph at the point (0, 0).

8.8 Angles

The author believes that time is well spent in considering the question:

What is an angle?

We observe, up front, that, on the surface, there are two relevant ques-tions here: (1) What is an angle? and (2) What is the measure of anangle?

When viewing a triangle drawn on a piece of paper, we sometimesplace a symbol, e.g. , inside the triangle near a vertex and then go onto speak of the angle . Somehow, human brains are wired closelyenough alike that the picture with its symbols can be used to developa general understanding about angles, even if the notion of angle has

not been fully defined.

We are fortunate that this communication is possible, for there dontappear to be words in the spoken language that can be used to define,in a non-technical manner, what the person on the street thinks of asan angle.

Angles, as we commonly think of them, are associated with two (pla-nar) chords, that share a common end point. The words the anglebetween them or the angle determined by them refer to an un-defined relationship between the two chords. So the question: What

is an angle? slips into the question of how to define the appropriaterelationship between two chords that share a common end point.

There are two geometric approaches that come to mind.

One can imagine a disc of radius one, with center at the commonend point of the two chords. The two half lines defined by the twochords contain chords of unit length that share the common end


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64 CHAPTER 8. EUCLIDEAN SPACES

point. These unit chords define two pie-shaped pieces of the unitdisc. That is, they define subsets of the Euclidean plane each ofwhich has a computable area (at least in principle). The subsetwith the smallest area can be defined to be the angle betweenthe two original chords. This definition results in an answer tothe question: What is an angle? The definition starts with thewords

An angle between two chords is a subset of the unit disc that

The measure of an angle so defined, can be taken to be twice thearea of the pie-shaped subset.

One can imagine the two chords above, each of unit length, inter-secting the boundary of the unit disc, i.e., the unit circle. Thesetwo intersection points determine two arcs of the unit circle. Thearc of smallest length can be defined to be the angle between thetwo original chords. This construction also results in a definitionthat starts with the words

An angle between two chords is an arc of the unit circle that

The measure of an angle so defined, can be taken to be the lengthof the chosen arc.

Looking back on these two ideas we conclude that they provide goodinsight, but mathematically speaking, they provide too much. Thenotion of angle, as it is used in practice, almost never uses the idea: anangle is a point set in R2. What is almost always used is: a measureof some relationship between two chords. (The relationship, whateverit is, is geometrically undefined, but garners general agreement.) So,

we settle for the idea that what is really needed in order to define thegeometric notion of angle between two chords, is just a real number.This is philosophically satisfying: an angle can be thought of as thevalue of a real-valued function defined on all pairs of non-degeneratechords that share an end point. What is the real-valued function?

We build on the notion that the measure of an angle can be taken tobe the length of an arc on the unit circle. The formal definition of the


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8.8. ANGLES 65

angle between two chords, given next uses vector space constructs. Itmodels the notion of an angle between two chords as discussed above.

Let x and y denote two non-zero vectors in R2. The angle betweenx and y is the real number given by

arccos

x, y||x||||y||

.

After defining the notion of arc length, it is possible to argue that thisdefinition of angle is exactly the length of the above-discussed arc onthe unit circle.

However, one observes that the definition of an angle exists withoutany notion of chords in the plane; it exists as an abstract constructin an inner-product space. For example this defines an angle betweentwo real-valued functions defined on the same real interval. Of courseits power in most geometric modeling rests on the use of 2-d and 3-dvectors to model chords in the plane and in 3-space.


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66 CHAPTER 8. EUCLIDEAN SPACES


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Chapter 9

ABSTRACT VECTOR

SPACES

The vector space structure ofRn, introduced in the previous chapter,can be applied to sets other than Rn. We introduce the abstractionbelow and call it a vector space. Similarly the idea of the distancebetween two points (vectors) in Rn can be abstracted. It is also in-troduced below and called a norm. Finally the notion of the innerproduct of two vectors in Rn can be abstracted. It is also introduced

below and called an inner product.

9.1 Definition of a Vector Space

A vector space over a field, F, is a non-empty abstract set, V,accompanied by two functions satisfying certain axioms. One func-tion maps V V into V and is called vector addition. The image of(v, w) V V, under vector addition, is denoted by v + w. Thevector, v + w, is called the sum of v and w. The other functionmaps F V into V and is called scalar multiplication. The image of(, v)

F V, under scalar multiplication, is denoted by v. The

vector, v, is called a scalar multiple of v.

The two functions are constrained by the following axioms.

(i) ( u, v, w V) (u + v) + w = u + (v + w).(additive associative law)

(ii) ( o V)( v V) o + v = v . (additive identity)

67


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68 CHAPTER 9. ABSTRACT VECTOR SPACES

(iii) (v V)( v V) v + (v) = o. (additive inverse)

(iv) ( v, w V) v + w = w + v. (additive commutative law)

(v) ( , F)( v V) ()v = (v). (scalar associative law)

(vi) ( F)( v, w V) (v + w) = v + w. (distributive law)

(vii) ( , F)( v V) ( + )v = v + v. (distributive law)

(viii) (v V) 1v = v.1 (scalar identity has a dual role)

Elements ofV are called vectors and elements ofFare called scalars.The field F is sometimes call the scalar field.2

With regard to the axioms of a vector space the following notes areentered.

Axiom (i) is a statement of the associative law for vector addi-tion. It permits one to write u + v + w without ambiguity.

The vector o in axiom (ii) is called the zero of the vector space,V. It is also called the additive identity. (One can prove that itis unique.)

The vector v in axiom (iii) is called the additive inverse of thevector v. (One can prove that it is unique.)

Axioms (vi) and (vii) are compatibility relations between vectoraddition and scalar multiplication. They are both distributivelaws.

1This is an innocuous looking axiom. One wonders whether or not it is im-plied by the other axioms. The other axioms prescribe a reasonably rich algebraicstructure. From them one can prove the following:

0v = o,

1o = o,

1(1v) = 1v, (This is almost axiom (vii)!) v = w implies v = w, (1v) = 1v.

However, without axiom (viii) one could define 1v = o, v V. All of the otheraxioms are consistent with this definition. The result is a trivial pseudo vectorspace.

2In these notes boldface symbols designate vectors unless otherwise noted.


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9.2. DEFINITION OF AN INNER PRODUCT 69

Axiom (viii) states that the multiplicative identity in the field,F, plays the role of a scalar multiplication identity.

The following observations are relevant.

The notion of multiplication of two vectors has not been defined.That is, there is nothing hypothesized about a function fromVV into V that behaves as a product of two vectors. Althoughthe notion of a cross product is widely used in the familiar vectorspace, R3, it plays no role, within the confines of abstract vector

spaces.

The expression v where v represents a vector and representsa scalar, has been assigned no meaning.

The cartesian product sets R2, R3, and in general Rn, n =2, 3, are vector spaces over R. Further, R is a vector spaceover itself.

A non-empty subset of a vector space is a subspace of the vectorspace if it is closed under addition and scalar multiplication.3

9.2 Definition of an Inner Product

Let V denote a vector space over the real field R. An inner producton V, denoted by v, w, for all v, w V, is a map from V V intoR that satisfies the following axioms:4

(i) (v V) v, v 0, (positivity)(ii) (v V) v, v = 0 v = o, (definiteness)

(iii) (u, v, w V)u + v, w = u, w + v, w , (distributive)(iv) (v, w V)( R) v, w = v, w, (homogeneity)(v) (v, w V) v, w = w, v. (symmetric)

An inner-product space is a vector space V over R on which aninner product has been defined.

3A subspace of a vector space is a vector space under the addition and scalarmultiplication it inherits for the (parent) vector space.

4Special considerations are needed when the real field R is replaced by thecomplex field C. We dont address the issue here.


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9.3 Definition of a Norm

Let V denote a vector space over the real field R. A norm on V is afunction, denoted by | | | |, from V into R that satisfies the followingaxioms:

(i) (v V) ||v|| 0, (positivity)

(ii) (v V) ||v|| = 0 v = o, (definiteness)

(iii) (

v

V)(

R)

||v

||=

|

| ||v

||, (homogeneity)

(iv) (v, w V) ||v + w|| ||v|| + ||w||. (triangle inequality)

A vector space upon which a norm is defined is called a normed vec-tor space (or a normed linear space).

Let u, v, w denote vectors in a normed vector space, over the real fieldR. The distance between u and v is defined to be u v. The def-inition of a norm ensures that: (1) the distance between u and v isa non-negative real number; (2) the distance between u and v is zeroif and only if u = v; (3) the distance between v and o is times

the distance between v and o, for > 0, (4) the distance between uand v is equal to or less than the distance between u and w plus thedistance between w and v.

For each p R, p 1, one can denote a norm on R2 by p and define it by

||(x1, x2)||p = (|x1|p + |x2|p)1p , (x1, x2) R2.

Further, one defines||(x1, x2)|| = max{|x1|, |x2|}, (x1, x2) R2.

One can argue thatlimp ||(x1, x2)||p = ||(x1, x2)||, (x1, x2) R2.

These are the so-called p-norms (-norm). Verifying that theabove does define a norm for each such p 1 is accomplished us-ing some special inequalities. That is not done here. The copies of thevector space, R2, endowed with these norms are called p-spaces.


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9.3. DEFINITION OF A NORM 71

Replacing R2 by Rn, n = 3, 4, in the definition of p-norms andreplacing (x1, x2) by (x1, x2, , xn) results in a definition ofp-normson Rn.

Let V denote an inner-product space and let||v|| = v, v, v V.

The norm, so defined, is said to be induced by the inner product.

Verifying that

v, v does indeed define a norm on V is left for the

Workbook.


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Chapter 10

METRIC SPACES

Above we have discussed the notion of distance in two instances: (i)the distance between two real numbers, given by the absolute value oftheir difference and (ii) the distance between two vectors in a normedvector space, given by the norm of their difference.1 In both of thesecases the distance was defined in terms of the underlying algebraicstructure (the difference of two vectors). History has shown that analgebraic structure is not needed in order to define a meaningful no-tion of distance between two elements in an abstract set. This idea

is introduced through the definitions of a metric and a metric space,given next.

10.1 Definition of a Metric

Let M denote a non-empty set. A function d : M M R is calleda metric on M if it satisfies the following axioms:2

(i) (x, y M) d(x, y) 0 with equality if and only if x = y,(ii) (

x, y

M) d(x, y) = d(y, x),

(ii) (x , y , z M) d(x, y) d(x, z) + d(z, y).

1Actually the two are the same in R since the absolute value function defines anorm on the (vector space of) real numbers.

2In the axioms we write d(x, y) when what is actually meant is d((x, y)) (thefunction d is being applied to the element (x, y) of M M). Omission of the insideparentheses is consistent with the literature; it makes an easier read.

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74 CHAPTER 10. METRIC SPACES

Let M denote a non-empty set and let d denote a metric on M. Theordered pair (M, d) is called a metric space.

The metric is frequently called the distance function, or just thedistance. It is common to speak of the metric space, M, usingonly the symbol for the set and omitting the symbol for the metric.

A short argument shows that the norm in a normed vector space isa metric on the underlying vector space. However, the notion of ametric exists independently of

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Fundamentals of Mathematics by Harry McLaughlin